Our computations in Magma computer algebra system [1]} show that many linear cyclic codes
satisfying the conditions of Proposition \(\ref {fix-subcode order 2}\)  part 1 have the same minimum distance as their fixed subcode by an order 2 multiplier.
In particular, for any \(a \in \mathbb {Z}_n\)  such that \({\rm ord}_n(a)=2\) , we computed the minimum distance of all non-trivial length \(n\)  linear cyclic codes with \(9<n<85\)  over \(\mathbb {F}_4\)  satisfying the conditions of Proposition  part 1. Among 72417 non-trivial such linear cyclic codes, 70256 of them have the same minimum distance as their corresponding fixed subcode by the action of \(\mu _{a}\) . The equality rate is about \(97\%\)  for all these codes.
In general, determining when a linear cyclic code and its fixed subcode by \(\mu _{-1}\)  have the same minimum distance appears
to be an interesting and presumably a difficult question.
